Transcript ENT 163 07-08 - UniMAP Portal

FUNDAMENTALS OF ELECTRICAL
ENGINEERING
[ ENT 163 ]
LECTURE #7
INTRODUCTION TO AC CIRCUITS
HASIMAH ALI
Programme of Mechatronics,
School of Mechatronics Engineering, UniMAP.
Email: [email protected]
CONTENTS
•
•
•
•
•
Introduction
Sinusoids
Phasors
Impedance and Admittance
AC Power Analysis
INTRODUCTION
 Dc circuit- excited by constant or time-invariant sources.
 Historically, dc sources were the main means of providing electric power
till the late 1800s.
 End of century – started to introduce an ac.
 Due to the advantages (more efficient and economical to transmit over
long distance) ac system is accepted.
SINUSOIDS
A sinusoid is a signal that has the form of the sine or cosine function.
 A sinusoid current is referred to as alternating current (ac). Such a
current reverse at regular time intervals and has alternately positive and
negative values.
 Circuits driven by sinusoidal current or voltage sources are called ac
circuits.
 Consider the sinusoidal voltage:
v(t )  Vm sin t
Vm= the amplitude of the sinusoid,
ω= the angular frequency in rads/s
ωt= the argument of the sinusoid
SINUSOIDS
 The sinusoid repeats itself every T seconds; thus T is called the period
of the sinusoid.
 Period; T is the required for a given sine wave to complete one full
cycle.
T
2

T  2
 Frequency f is the number of cycles that a sine
wave complete in one second
1
f 
T
  2f
SINUSOIDS
 Peak to peak Voltage Vp-p: is the value of twice the peak voltage (amplitude).
 The root-mean-square (RMS) value of a sinusoidal voltage is equal to the
dc voltage that produces the same amount of heat in a resistance as does
the sinusoidal voltage.
Vrms
Vm

2
or
Vrms  0.707V p
or
V p  1.414Vrms
 The average value of a sine wave – defined over a half cycle of the wave.
Varg  0.6371V p
SINUSOIDS
Let consider a more general expression for the sinusoid:
v(t )  Vm sin( t   )
Where

is the phase
Vm
v1 (t )  Vm sin t

-Vm
v2 (t )  Vm sin( t   )
SINUSOIDS
 Therefore we say that v2 leads v1 by
 For
 If
 0
 0

or v1 lag v2 by  .
we could also say that v1 and v2 are out of phase.
, then v1 and v2 are said to be in phase.
Example:
Find the amplitude, phase, period and frequency of the sinusoid
v(t )  12 sin( 50t  10)
SINUSOIDS
Example:
Calculate the phase angle between v
1
(t )  10 cos(t  50) and
v(t )  12 sin( t  10) . State which sinusoid is leading.
PHASORS
A phasor is a complex number that represents the amplitude and phase of a
sinusoid.
Three representation of complex number:
 Rectangular form:
 Polar form:
z  x  jy
z  r
 Exponential form:
z  re j
j=(√-1),
or
z  x  jy  r  r (cos   j sin  )
x= real part of z,
y=imaginary part of z,
r=magnitude of z,
Ø=phase of z.
PHASORS
 Important operations involving complex numbers:
 Addition
z1  z2  ( x1  x 2 )  j( y1  y2 )
 Subtraction
z1  z2  ( x  x 2 )  j ( y1  y2 )
 Multiplication
z1 z2  r1r 2 (1  2 )
 Division
z1 r1
 (1  2)
z 2 r2
 Reciprocal
1 1
1
 ( 1),   j
z r
j
PHASORS
Square root
z  r ( / 2)
Complex conjugate
z   x  jy  r( )  re  j
Phasor representation
v(t )  Re(Ve jt ), V  Vm e j  Vm 
PHASORS
Representation of
Ve jt
PHASORS
Phasor diagram:
 The phasors are rotating anticlockwise as indicated by the arrowed
circle. A is leading B by 90 degrees.
 Since the two voltages are 90 degrees apart, then the resultant can be
found by using Pythagoras, as shown.
.
PHASORS
Transformation of sinusoid from the time domain to phasor domain
v(t )  Vm cos(t   )
V  Vm
Sinusoid-phasor transformation
Time-domain
Phase-domain
Vm cos(t   )
Vm 
Vm sin( t   )
Vm   90
I m cos(t   )
I m 
I m sin( t   )
I m   90
PHASORS
•
Phasor-domain= frequency domain
•
Differences between v(t) and V:
• v(t) is the instantaneous or time-domain representation, while
V is the frequency or phasor-domain representation.
• v(t) is time dependent, while V is not.
• v(t) is always real with no complex term, while V is generally
complex
PHASORS RELATIONSHIPS FOR CIRCUIT
ELEMENTS
1. Resistor
•
If the current through a resistor R is
voltage,
v  Ri  RI m cos(t   )
V  R
Phasor form
i  I m cos(t   ) then, the
V  RI m
  I m 
PHASORS
Phasor Relationships for Circuit Elements.
Element
R
Time-domain
representation
v  Ri
Phasor-domain
representation
V  RI
L
vL
di
dt
V  jRi
C
iC
dv
dt
I  jCV
IMPEDANCE AND ADMITTANCE
•
Previously, we know that the voltage-current relations for the three
passive elements as:
V=IR,
•
V  jRi
V/I= 1/JωC
From here, we obtain Ohm’s Law in phasor as
Z=V/I
•
I  jCV
The above equation can be written in terms of the ratio of phasor
voltage to phasor current as:
V/I=R V/I= JωL
•
,
V=ZI
Impedance, Z is the ratio of the phasor voltage V to the phasor current
I, measured in ohms (Ω)
IMPEDANCE AND ADMITTANCE
Element
Impedance
Admittance
R
Z=R
Y=1/R
C
Z=jωL
Y=1/jωL
L
Z=1/jωC
Y=jωC
•
When ω=0 (dc source), ZL=0 and Zc=∞ (confirm- inductor acts like short
circuit, capacitors acts like open circuit).
•
When ω=∞ (high frequencies ) ZL= ∞ and Zc=0 (indicate- inductor acts
like open circuit, capacitors acts like short circuit).
Further Reading
Fundamentals of electric circuit. (2th Edition), Alexander, Sadiku, MagrawHill.
(chapter 9).